AC Power & Waveforms

The full-rigor companion to the power triangle: enter a load as R-L-C elements, an impedance, or P and power factor, then shape the current with any harmonic spectrum — presets or order-by-order magnitude and phase — and get the complete IEEE 1459 picture: P, Q₁, distortion power D, apparent S, true vs. displacement power factor, and THD, with a live one-line, waveforms, spectrum, phasors, and the power tetrahedron. One scenario — every view updates together. Drive-specific spectra and their mitigation ladder live on the VFD Harmonics page.

Source
V
Assumed sinusoidal — see Model scope.
Load
kW
cos θ
Current Harmonics
Order nIₙ/I₁ (%)Phase (°)

Percent of the fundamental current. Presets are idealized/typical spectra — verify against the actual equipment's harmonic data. Enter each harmonic order once — components at the same frequency must be phasor-combined before entry. With one row per order, phase angles shape the waveform and its peak but do not change RMS, THD, or the power numbers (with sinusoidal voltage).

One-Line
Waveforms — v(t), i(t), and p(t) = v·i
v(t) i(t) p(t) Average of p(t) = P

Harmonic Spectrum — current

Fundamental Phasors

Lengths normalized per quantity; magnitudes in the readout below.

Power Triangle / Tetrahedron

Readout — per phase
Balanced Three-Phase Totals

3 × per-phase with VLL = √3·VLN — balanced systems only.

Method — your values substituted
Equations

Time-varying quantities

The harmonic-sum RMS applies to any periodic waveform; Vₙ are RMS magnitudes.

Element impedances

At harmonic n, XL scales by n and XC by 1/n — why inductors starve and capacitors attract harmonic current.

Complex power

Power under distortion (IEEE 1459)

With sinusoidal voltage. Capacitor banks correct displacement PF only — they do nothing for the distortion term (and may resonate with it — see RLC resonance).

Δ-Y relations (balanced, ABC rotation)

Model Scope
  • Source voltage is sinusoidal. Harmonics are applied to the current only — the stiff-bus, nonlinear-load case. With sinusoidal voltage, only the fundamental current carries real power; voltage distortion and system-impedance interaction are out of scope.
  • Harmonic presets are idealized/typical spectra — the 1/n six-pulse spectrum is an upper-bound idealization. Verify against the actual equipment's harmonic data (drive datasheet, measurement).
  • Three-phase totals assume a balanced system (3× per-phase; VLL = √3·VLN).
  • Steady state, single fundamental frequency. Q under distortion is reported as fundamental reactive power Q₁ (IEEE 1459); other Q definitions exist for nonsinusoidal conditions.
  • This page reports exact quantities for the stated load model — it does not size equipment or screen against IEEE 519 harmonic limits.
Validation — recomputed live at load
CaseExpectedComputedResult
References
  1. J. D. Glover, T. J. Overbye, and M. S. Sarma, Power System Analysis & Design, 6th ed., Cengage, 2017, ch. 2.
  2. IEEE Std 1459-2010, Standard Definitions for the Measurement of Electric Power Quantities Under Sinusoidal, Nonsinusoidal, Balanced, or Unbalanced Conditions.
  3. IEEE Std 519-2022, Standard for Harmonic Control in Electric Power Systems (THD definition).